Using the Bockstein to show that $\mathbb{R}P^3/\mathbb{R}P^1$ and $\Sigma \mathbb{R}P^2$ are not homotopy equivalent. The previous exercise asked to show that the cohomology rings $H^*(\Sigma \mathbb{R}P^2; \mathbb{Z}/2)$ and $H^*(\mathbb{R}P^3/\mathbb{R}P^1; \mathbb{Z}/2)$ are isomorphic, so we should probably take the Bockstein associated to
$$\mathbb{Z}/2 \to \mathbb{Z}/4 \to \mathbb{Z}/2$$
The cohomology groups are $\mathbb{Z}/2, 0, \mathbb{Z}/2, \mathbb{Z}/2, 0, 0, ...$. As a homotopy equivalence between spaces induces an isomorphism on cohomology groups it suffices to show that the only non-trivial Bockstein from second to third cohomology has a different action. 
We can use the universal coefficients theorem to calculate $H^2(\Sigma \mathbb{R}P^2; \mathbb{Z}/4) \cong H^1(\mathbb{R}P^2; \mathbb{Z}/4) \cong \mathbb{Z}/2$, $H^2(\mathbb{R}P^3/\mathbb{R}P^1; \mathbb{Z}/4) \cong 0$, $H^3(\Sigma \mathbb{R}P^2; \mathbb{Z}/4) \cong H^2(\mathbb{R}P^2; \mathbb{Z}/4) \cong (\mathbb{Z}/4)/2 \cong \mathbb{Z}/2$ and $H^3(\mathbb{R}P^3/\mathbb{R}P^1; \mathbb{Z}/4) \cong \mathbb{Z}/4$. As the second homology with $\mathbb{Z}/4$ coefficients differ, we can already conclude that the spaces are not homotopy equivalent.
So we finally detect a difference. The long exact sequence associated to $\Sigma \mathbb{R}P^2$ is 
$$\mathbb{Z}/2 \to \mathbb{Z}/2 \to \mathbb{Z}/2 \xrightarrow{\beta} \mathbb{Z}/2 \to \mathbb{Z}/2 $$
and the one associated to $\mathbb{R}P^3/\mathbb{R}P^1$ is
$$\mathbb{Z}/2 \to 0 \to \mathbb{Z}/2 \xrightarrow{\beta} \mathbb{Z}/2 \to \mathbb{Z}/4$$
So the second $\beta$ must be the identity, while the first one could possibly be zero.
Source:

 A: Since this is an exercise for a course, I will try to give hints rather than a complete answer. I don't know what you've covered in your course so far, so hopefully you're able to fill in the gaps with your available material (the proofs I have in mind use relatively basic properties). In this answer all cohomology groups are assumed to have $\mathbb{Z}/2$ coefficients.
The Bockstein is a stable cohomology operation, meaning that if $\sigma \colon \tilde{H}^*(X) \to \tilde{H}^{*+1}(\Sigma X)$ is the suspension isomorphism then we have the following commutative diagram: 
$\require{AMScd}$
\begin{CD}
H^n(X) @>{\beta}>> H^{n+1}(X)\\
@V{\sigma}VV @V{\sigma}VV\\
H^{n+1}(\Sigma X) @>{\beta}>> H^{n+2}(\Sigma X)
\end{CD}
i.e. $\beta\circ \sigma = \sigma \circ \beta$. Use stability to compute $\beta \colon H^2(\Sigma \mathbb{RP}^2) \to H^3(\Sigma \mathbb{RP}^2)$.
It is also natural, in particular the quotient map $q\colon \mathbb{RP}^3 \to \mathbb{RP}^3/\mathbb{RP}^1$ gives a commutative diagram
$\require{AMScd}$
\begin{CD}
H^2(\mathbb{RP}^3/\mathbb{RP}^1) @>{\beta}>> H^3(\mathbb{RP}^3/\mathbb{RP}^1)\\
@V{q^*}VV @V{q^*}VV\\
H^2(\mathbb{RP}^3) @>{\beta}>> H^3(\mathbb{RP}^3)
\end{CD}
Note that the vertical maps are isomorphisms by the long exact sequence of the pair. Use this diagram to help compute $\beta \colon H^2(\mathbb{RP}^3/\mathbb{RP}^1) \to H^3(\mathbb{RP}^3/\mathbb{RP}^1)$.
